**Models of PA**

** Wednesday, February 27, 2013 6:45 pm**

Speaker: Erez Shochat St. Francis College

Title: Introduction to interstices and intersticial gaps II

**Set theory seminar**

** Friday, March 1, 2013 10:00 am**

Speaker: Joel David Hamkins The City University of New York

Title: The omega one of chess

This talk will be based on my recent paper with C. D. A. Evans,

Transfinite game values in infinite chess.

Infinite chess is chess played on an infinite chessboard. Since

checkmate, when it occurs, does so after finitely many moves, this is

technically what is known as an open game, and is therefore subject to

the theory of open games, including the theory of ordinal game values.

In this talk, I will give a general introduction to the theory of

ordinal game values for ordinal games, before diving into several

examples illustrating high transfinite game values in infinite chess.

The supremum of these values is the omega one of chess, denoted by

$omega_1^{mathfrak{Ch}}$ in the context of finite positions and by

$omega_1^{mathfrak{Ch}_{hskip-2ex atopsim}}$ in the context of all

positions, including those with infinitely many pieces. For lower

bounds, we have specific positions with transfinite game values of

$omega$, $omega^2$, $omega^2cdot k$ and $omega^3$. By embedding trees

into chess, we show that there is a computable infinite chess position

that is a win for white if the players are required to play according

to a deterministic computable strategy, but which is a draw without

that restriction. Finally, we prove that every countable ordinal

arises as the game value of a position in infinite three-dimensional

chess, and consequently the omega one of infinite three-dimensional

chess is as large as it can be, namely, true $omega_1$.

**Model theory seminar**

** Friday, March 1, 2013 12:30 pm**

Speaker: Alf Dolich The City University of New York

Title: Maximal Automorphisms

A maximal automorphism of a structure M is an automorphism under which

no non-algebraic element of M is fixed. A problem which has attracted

some attention is when for a theory T any countable recursively

saturated model of T has a maximal automorphism. In this talk I will

review what is known about this problem in various contexts and then

prove a general result that guarantees, under certain mild conditions

on T, that any countable recursively saturated model of T does indeed

have a maximal automorphism.

**CUNY Logic Workshop**

** Friday, March 1, 2013 2:00 pm**

Speaker: Tin Lok Wong Ghent University

Title: Understanding genericity for cuts

In a nonstandard model of arithmetic, initial segments with no maximum

elements are traditionally called cuts. It is known that even if we

restrict our attention to cuts that are closed under a fixed family of

functions (e.g., multiplication, the primitive recursive functions, or

the Skolem functions), the properties of cuts can still vary greatly.

I will talk about what genericity means amongst such great variety.

This notion of genericity comes from a version of model theoretic

forcing devised by Richard Kaye in his 2008 paper. Some ideas were

already implicit in the work by Laurence Kirby and Jeff Paris on

indicators in the 1970s.

**Computational Logic Seminar**

** Tuesday, March 5, 2013 2:00 pm**

Speaker: Sergei Artemov The CUNY Graduate Center

Title: Lost in translation: a critical view of epistemic puzzles solutions.

There are two basic ways to specify an epistemic scenario:

1. Syntactic: a verbal description with some epistemic logic on the

background and some additional formalizable assumptions; think of the

Muddy Children puzzle.

2. Semantic: providing an epistemic model; think of Aumann structures

– a typical way to define epistemic components of games.

Such classical examples as Muddy Children, Wise Men, Wise Girls, etc.,

are given by syntactic descriptions (type 1), each of which is

“automatically” replaced by a simple finite model (type 2). Validity

of such translations from (1) to (2) will be the subject of our study.

We argue that in reducing (1) to (2), it is imperative to check

whether (1) is complete with respect to (2) without which solutions

of puzzles by model reasoning in (2) are not complete, at best. We

have already shown that such reductions can be justified in the Muddy

Children puzzle MC due to its model categoricity: we have proved that

MC has the unique “cube” model Q_n for each n. This fixes an obvious

gap in the “textbook” solution of Muddy Children which did not provide

a sufficient justification for using Q_n.

We also show that an adequate reduction of (1) to (2) is rather a

lucky exception, which makes the requirement to check the completeness

of (1) w.r.t. (2) necessary. To make this point, we provide a

simplified version of Muddy Children (by dropping the assumption “no

kid knows whether he is muddy”) which admits the usual deductive

solution by reasoning from the syntactic description, but which cannot

be reduced to any finite model.

**Models of PA**

** Wednesday, March 6, 2013 6:45 pm**

Speaker: Keita Yokoyama Mathematical Institute, Tohoku University

Title: Several versions of self-embedding theorem

In this talk, I will give several versions of Friedman’s

self-embedding theorem which can characterize subsystems of Peano

arithmetic. Similarly, I will also give several variations of Tanaka’s

self-embedding theorem to characterize subsystems of second-order

arithmetic.

**Set theory seminar**

** Friday, March 8, 2013 10:00 am**

Speaker: Robert Lubarsky Florida Atlantic University

Title: Forcing for Constructive Set Theory

As is well known, forcing is the same as Boolean-valued models. If

instead of a Boolean algebra one used a Heyting algebra, the result is

a Heyting-valued model. The result then typically models only

constructive logic and falsifies Excluded Middle. On the one hand,

many of the same intuitions from forcing carry over, while on the

other the result is quite foreign to a classical mathematician. I will

give a survey of perhaps too many examples, and call for the

importation of more methods from current classical set-theory into

constructivism.

**CUNY Logic Workshop**

** Friday, March 8, 2013 2:00 pm**

Speaker: Keita Yokoyama Mathematical Institute, Tohoku University

Title: Reverse mathematics for second-order categoricity theorem

Link: http://nylogic.org/talks/title-tba-4

It is important in the foundations of mathematics that the natural

number system is characterizable as a system of 0 and a successor

function by second-order logic. In other words, the following

Dedekind’s second-order categoricity theorem holds: every Peano system

$(P,e,F)$ is isomorphic to the natural number system $(N,0,S)$. In

this talk, I will investigate Dedekind’s theorem and other similar

statements. We will first do reverse mathematics over $RCA_0$, and

then weaken the base theory. This is a joint work with Stephen G.

Simpson.

http://nylogic.org/talks/lost-in-translation-a-critical-view-of-epistemic-puzzles-solutions